To put a simpler spin on this, the answer is: Yes (although you really mean the JVM rather than the JRE). The program that the OS is running is the JVM (Java virtual machine), and the Java application is data being read by that program. The JVM is like Microsoft Word, and Java programs are like Word documents.
This question is hitting upon the essential difference between compiled and interpreted languages, as described well here.
To use the analogy further to explain what the JVM and JRE are…The JVM is like the Microsoft Word program itself, and the JRE is like the MS Word program plus all the other stuff, like templates, sample documents, fonts, etc. that is installed along with it to support what it does.